ИСТИНА

There are (at least) two different approaches to define equivariant analogue of the Euler charateristic for spaces with a finite group actions. The first one defines the equivariant Euler characteristic of a space with a finite group action as an element of the Burnside ring of the group. The second approach emerged from physics and includes the so-called orbifold Euler characteristic and its higher order versions. These Euler characteristics are integers. Both approaches can be extended to generalized (universal) Euler characteristics with values in appropriate Grothendieck rings of quasiprojective varieties (or rather in their modifications). A Macdonald type equations for an invariant is an equation for the generating series of the invariant for the symmetric powers $S^nX$ of a space $X$ (or for their analogues) describing it as an exponent with the base not depending on the space and the power equal to the value of the invariant for the space itself. E.g., for the usual Euler characteristic $\chi$ the Macdonald type equation has the form $$1+\sum_{n=1}^{\infty}\chi(S^nX)t^n=(1-t)^{\chi(X)}\,.$$ We give a way to merge the two described approaches together defining (in the situation with two commuting finite group actions) higher order (orbifold) Euler characteristics with values in the Burnside ring of a group. We give Macdonald type equations for these invariants. We also offer generalized ("motivic") versions of these invariants with values in modifications of appropriate Grothendieck rings of quasiprojective varieties and formulate Macdonald type equations for them as well. The latter ones are formulated in terms of the so-called power structure over the Grothendieck ring of quasiprojective varieties defined earlier. The talk is based on a joint work with I.Luengo and A.Melle-Hernández.